Integrand size = 24, antiderivative size = 278 \[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=-\frac {2 (8 c d-11 b e) x \sqrt {d+e x^3}}{55 e^2}+\frac {2 c x^4 \sqrt {d+e x^3}}{11 e}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2-22 b d e+55 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \]
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Time = 0.14 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1425, 396, 224} \[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (55 a e^2-22 b d e+16 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}-\frac {2 x \sqrt {d+e x^3} (8 c d-11 b e)}{55 e^2}+\frac {2 c x^4 \sqrt {d+e x^3}}{11 e} \]
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Rule 224
Rule 396
Rule 1425
Rubi steps \begin{align*} \text {integral}& = \frac {2 c x^4 \sqrt {d+e x^3}}{11 e}+\frac {2 \int \frac {\frac {11 a e}{2}-\left (4 c d-\frac {11 b e}{2}\right ) x^3}{\sqrt {d+e x^3}} \, dx}{11 e} \\ & = -\frac {2 (8 c d-11 b e) x \sqrt {d+e x^3}}{55 e^2}+\frac {2 c x^4 \sqrt {d+e x^3}}{11 e}-\frac {1}{55} \left (-55 a-\frac {2 d (8 c d-11 b e)}{e^2}\right ) \int \frac {1}{\sqrt {d+e x^3}} \, dx \\ & = -\frac {2 (8 c d-11 b e) x \sqrt {d+e x^3}}{55 e^2}+\frac {2 c x^4 \sqrt {d+e x^3}}{11 e}+\frac {2 \sqrt {2+\sqrt {3}} \left (16 c d^2-22 b d e+55 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right )|-7-4 \sqrt {3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.35 \[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\frac {x \left (-2 \left (d+e x^3\right ) \left (8 c d-11 b e-5 c e x^3\right )+\left (16 c d^2+11 e (-2 b d+5 a e)\right ) \sqrt {1+\frac {e x^3}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {e x^3}{d}\right )\right )}{55 e^2 \sqrt {d+e x^3}} \]
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Time = 0.86 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {2 x \left (5 c \,x^{3} e +11 b e -8 c d \right ) \sqrt {e \,x^{3}+d}}{55 e^{2}}-\frac {2 i \left (55 a \,e^{2}-22 b d e +16 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{165 e^{3} \sqrt {e \,x^{3}+d}}\) | \(333\) |
elliptic | \(\frac {2 c \,x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (b -\frac {8 c d}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}-\frac {2 i \left (a -\frac {2 d \left (b -\frac {8 c d}{11 e}\right )}{5 e}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\) | \(343\) |
default | \(\text {Expression too large to display}\) | \(907\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.26 \[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\frac {2 \, {\left ({\left (16 \, c d^{2} - 22 \, b d e + 55 \, a e^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) + {\left (5 \, c e^{2} x^{4} - {\left (8 \, c d e - 11 \, b e^{2}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{55 \, e^{3}} \]
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Time = 1.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.43 \[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\frac {a x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {4}{3}\right )} + \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {7}{3}\right )} + \frac {c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {10}{3}\right )} \]
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\[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{\sqrt {e x^{3} + d}} \,d x } \]
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\[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{\sqrt {e x^{3} + d}} \,d x } \]
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Timed out. \[ \int \frac {a+b x^3+c x^6}{\sqrt {d+e x^3}} \, dx=\int \frac {c\,x^6+b\,x^3+a}{\sqrt {e\,x^3+d}} \,d x \]
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